A Generalization of the Correspondences Between Quasi-Hereditary Algebras and Directed Bocses

Quasi-hereditary algebras were introduced by Cline, Parshall and Scott to study the highest weight categories in Lie theory. On the other hand, bocses were introduced in the context of Drozd’s tame and wild dichotomy theorem. Koenig, Külshammer and Ovsienko connected the two areas by giving equivalences between the categories of \(\Delta \)-filtered modules over quasi-hereditary algebras and those of modules over directed bocses. In this article, we extend this result to \(\overline{\Delta }\)-filtered algebras. We face two problems when proving a similar theorem for \(\overline{\Delta }\)-filtered algebras. The first one is that the \(\textrm{Ext}\)-algebra of proper standard modules may be infinite dimensional. The second one is that the underlying algebra B of the bocs \(\mathcal {B}\) induced from a \(\overline{\Delta }\)-filtered algebra may be infinite dimensional. We give solutions for these problems and show the relationship between the categories of \(\overline{\Delta }\)-filtered modules over \(\overline{\Delta }\)-filtered algebras and those of modules over some class of bocses.